Four Dimensional Localisation With Motivic

Neutrinos

M. D. Sheppeard

Aranui, Christchurch 8061, New Zealand

E-mail: marnisheppeard@gmail.com

Abstract. Quantum gravity traditionally begins with path integrals for four dimensionalspacetimes, where the subtlety is in smooth structures. From a motivic perspective, the samediagrams belong to ribbon categories for quantum computation, based on algebraic numberfields. Here we investigate this divide using the principle of the neutrino CMB correspondence,which introduces a mirror pair of ribbon diagrams for each Standard Model state. Categoricalcondensation for gapped boundary systems in extended quantum double models extends themodular structure to encompass Kirby diagrams.

Keywords: algebraic integers, gapped boundary, modular tensor, motivic, neutrino

1. IntroductionIn motivic gravity we study the localisation to four dimensions from a categorical perspective,which is radically different from the traditional one. From this viewpoint [1][2], physical axiomsbecome increasingly complex, beyond the knot and ribbon categories [3][4][5] in dimension3, which underlie topological quantum computation for anyon and gapped boundary systems[6][7][8][9][10]. We permit additional time directions, since every string in a quantum circuitis permitted to twist in its time domain, prior to the consideration of a global spacetime. Insix dimensions, three times for mass generation are localised to a single time coordinate by theneutrino CMB correspondence [11][12], which associates one right handed neutrino mass to thepresent day CMB temperature. This thermal gravity is potentially evidenced by experimentson the topological thermal Hall effect [13][14].

In the gauge theory setting, the Jones polynomial appears [15] with Wilson loops for theChern-Simons action. Inspired by a holographic principle for a computational mirror, orMajorana holography [16], we study Witten’s tower of dimensions [17][18] using categoricalalgebra and number theory. The Jones invariant [19] is interpreted [17] in 4D using electricmagnetic duality, and this is extended to 5D with categorification for Khovanov homology[20][21].

Electric magnetic duality is associated to the dyonic structure of ribbon particle states, whichappear as mirror pairs. In the basic scheme, B3 diagrams in S3 acquire a U(1) ribbon twist to givean SU(2)× U(1) compactified Minkowski space as an emergent feature of discrete SU(2) braidgroup representations. Since the gauge groups determine the spacetime, a complexification ofthe Chern-Simons action immediately suggests a six dimensional setting. Motivic gravity startshere [2] with two copies of CSFT: one for QCD and one for the IR scale of neutrino mass [22][23].

Topology change is morally a topos theoretic concept [24]. Exotic structures on 4-manifoldsare defined by the integral form and associated knot and link diagrams [25]. In quantum topostheory, real analysis is not a good starting point for emergent geometries in low dimensions, andone expects the axioms to merge algebraic number theory and combinatorics, just as Feynmanamplitudes are evaluated in a ring of periods rather than C. The appendix shows how algebraicnumber theory is linked to basic observables in quantum mechanics. We insist that observablesdictate algebra, in order to minimise the analytical baggage that must be carried around, andanalysis should be re-axiomatised in quantum logic.

The associahedron [26][27] polytope is an axiom for an n-category. In this paper we explainhow the associahedra are closely related to knots. The connection between instanton moduliand knots is studied using fermion condensates [28], both for (supersymmetric) QCD and an IRcounterpart, which we attribute to neutrino gravity [2]. Knots are extended to ribbons, which arethe building blocks of template diagrams. All diagrams are interpreted category theoretically,focusing on Fibonacci anyon categories and condensation algebras [29], which we relate to theKirby calculus.

Section 2 introduces the golden rings, integral forms and electric charges. In section 3 we putelectric magnetic duality in a motivic framework, review the Koide rest mass phenomenology,and list the fundamental states of the Standard Model. Section 4 discusses categorical algebrasand polytopes, and section 5 concludes with comments on motivic integration. The essentialclaim is that mass generation in gapped boundary systems has an abstract analog under theneutrino CMB ansatz, which is the true origin of inertial mass in quantum gravity. Here we donot discuss Yang-Mills doubles, entanglement measures, black holes or AdS geometries, althoughobvious connections exist.

2. Towards 4-manifolds2.1. The golden ring and Fibonacci categoriesExperimental precision does not actually require R or C, except in axiomatic questions ofcomputability. Like the adeles, where R appears as the infinite prime, we imagine real manifoldsemerging in infinite dimensional computations, whereas a qudit state space makes do withalgebraic numbers, canonically selected by the quantum mechanical question.

Consider the various triangles in the pentagram of the appendix. The little blue trianglebelow the bisected top spike of the pentagram is equivalent to the top blue right angled triangle,with an angle of 18◦ = tan−1(φρ)−1, with φ = (1 +

√5)/2 the golden ratio and ρ =

√φ+ 2 the

diagonal of the golden rectangle. The little 18◦ bisects the 36◦ at the red chord, which is a pieceof a smaller red pentagon, initiating a discrete zoom-in quasilattice of pentagonal coordinatesfor the plane. In [30] it is shown that the 8 dimensional rational half integers Z8/2 may beembedded in C using the golden ring Z[ρ] (see the appendix). One basis of R ⊂ C is given by

x = x0 + x1φ+ x2ρ+ x3φρ (1)

for integral xi. Eight dimensions, in the form x+ iy, is an obvious setting for the e8 lattice andits intersection form.

There are four types [31] of Fibonacci ribbon category that use golden geometry: two based onminimal models and two for the affine chiral algebras G2,1 and F4,1. Note that G2,1×F4,1 ⊂ E8,1.The affine VOAs correspond to a central charge

c =k dim(g)

k + h∨(2)

at level k, and in general the deformation parameter u is given by

u−1 = eπic/2. (3)

For φ(u2 + 1) = −u, we have the usual Fibonacci objects I and X with the quantum dimensionof X equal to φ. The Yang-Lee model at the tenth root u = eπi/5 is the representations of themodel M(2, 5) with c = −22/5. In the modular categories, for a unitary VOA, the ribbon twistis given by the conformal weight. The modular matrices here take the form

S = ρ−1(

1 φφ −1

), T = u1/6

(1 00 u2

). (4)

For G2,1 at u = e3πi/5 the phase u1/6 in T is still golden, and similarly for F4,1.If we wanted a basic deformation of ω = (−1 +

√−3)/2 for the Eisenstein integers, we would

require c = 6, which appears in a (4, 4) superconformal theory for Mathieu moonshine.We do not use gauge theory to evaluate knot invariants. The Jones or HOMFLYPT

polynomials are evaluated using skein relations, and the Khovanov complex is defined as usualusing smoothings. Since our link strands are not geometric in the classical sense, what mattersis the information content, or complexity, of a diagram.

2.2. The role of e8A 4-manifold is characterised by its integral form [25], and a key component in the classificationof integral forms is the e8 form

E8 =

2 1 0 0 0 0 0 01 2 1 0 0 0 0 00 1 2 1 0 0 0 00 0 1 2 1 0 0 00 0 0 1 2 1 0 10 0 0 0 1 2 1 00 0 0 0 0 1 2 00 0 0 0 1 0 0 2

. (5)

But an e8 manifold cannot be smooth because the form does not diagonalise over the rationalintegers. It does, however, diagonalise [32] over the golden integers Z[φ], where φ = (1 +

√5)/2

is the golden ratio. As is well known, E8 is positive definite, even and unimodular for closedmanifolds. Let σ(Q) be the signature of the form Q and r(Q) its rank. All such quadratic formstake the form [25]

Q =σ(Q)

8E8 ⊕

r(Q)− |σ(Q)|2

(0 11 0

), (6)

where the 2×2 factors are forms for either the torus T 2 (for H1(T2,Z2)) or S2×S2 in dimension 4.

This ambiguity will matter later on. One important smooth space, which contains a topologicalcomponent that cannot be smoothed, is the Kummer (or K3) surface, with its 22 dimensionalform

QK3 = E8 ⊕ E8 ⊕(

0 11 0

)⊕(

0 11 0

)⊕(

0 11 0

). (7)

The construction uses a connected sum of three copies of the 4-manifold S2×S2, correspondingto the triple of σX flip matrices. A quotient of this sum by a certain embedding, relatedto the Kummer surface, yields a fake version of R4, wherein a theorem of Freedman [33]characterises good copies of R4 by the conditions: simply connected, non-compact, withoutboundary, H2(M,Z) = 0, homeomorphic to S3 × [0,∞). It turns out that there are anuncountable number of inequivalent smooth structures on R4, and this complexity is uniqueto dimension 4 due to surface knotting.

The elliptic genus for the K3 surface [34] is

ZK3(τ, z) = 8[(θ2(τ, z)

θ2(τ, 0))2 + (

θ3(τ, z)

θ3(τ, 0))2 + (

θ4(τ, z)

θ4(τ, 0))2], (8)

for θi(τ, z) the Jacobi theta functions. Nowadays it is written in terms of dimensions ofirreducible representations [35] for the Mathieu group M24, in Mathieu moonshine [36] andits umbral generalisation [37]. Moonshine for the Monster group uses the j-invariant, which isanalogously defined by

j(q) = 32(θ2(0, q)

8 + θ3(0, q)8 + θ4(0, q)

8)3

(θ2(0, q)θ3(0, q)θ4(0, q))8. (9)

The associated Eisenstein series E4 counts vectors in the shells of the e8 root lattice. Thelattices underlying modular forms are fundamental. Whenever we see a triplet of terms, as in(9), we think of some form of triality. In the Jordan algebra J3(O), triality acts on the threeoff-diagonal copies of O, and this extends to generalised 3× 3 algebras for ribbon categories.

How about the physics? As in table 1, the exact gauge symmetries U(1)Q and color SU(3)Care directly associated, respectively, to ribbon twists and a triplet of ribbon strands, so thatthe 8 negative electric charges for {ν, e−, d, u} lie on the vertices of a 3-cube [38][39], whosedirections label ribbon strands. Such a 3-cube, in figure 1, determines a basis for the octonions O.Complexification brings in antiparticles, for a total 4-cube of charges, and complex conjugationis charge conjugation in the C ⊗ O ideal algebra, which extends to crossing flips in the ribbonpicture. A 7-cube introduces magnetic data with 7-stranded ribbon diagrams, and in generalthe extended M theory dimension equals twice the number of ribbon strands.

Electroweak symmetries are broken back-to-front: the Higgs mass emerges from the inversesee-saw rule [2], which pairs the Planck scale and IR neutrino mass scale at 0.01 eV. There is noneed to introduce further local states beyond the SM, because the new RH neutrino is associatedto a cosmological scale.

2.3. The Kirby calculus and electric chargeAn axiomatic approach to invariants constructs state sums from categorical data. Barrett etal [40] consider a functor from a spherical fusion category into a ribbon fusion category for the2-handlebody of the 4-manifold, in order to define smooth invariants by weakening the slidinglaw in a ribbon category. A 4-manifold is determined up to diffeomorphism by its 2-handlebodyattachments, where a 2-handle piece is essentially a framed knot in the S3 boundary of the0-handle.

For a 2-handle in 4 dimensions, the framing classes are characterised by π1(O(2)) = Z [25],whereas in 3 dimensions the only framings are 1 or θ1/2, where θ represents a full ribbon twist.This distinction is crucial here, because quantised electric charge is given by ribbon twists.Three dimensions only permits two charges, whereas physics requires three values 0,±1. Oftenwe denote these charges [1] by the cubed roots of unity 1, ω and ω, when θ behaves like a phase.To obtain this Z3 we have to go to four dimensions, and quotient out by the natural 3Z: a tripleof half twists θ3/2 equals −1 under the cubed root representation, while B2 = Z represents −1by a single half twist. In other words, electric charge appears with the holographic extension of2 + 1 dimensions through the Kirby interpretation of diagrams.

A Kirby diagram [41] of knots and balls accounts for all the attachments in R3 for a 4-manifold(there are really no real numbers; just diagrams). A single knot in the Kirby diagram will cancela 1-handle (two balls in R3) if it has ends attached to each S2 boundary. In an Akbulut diagram,the balls of a 1-handle are replaced by marked circles, so that the entire 4-manifold is specified

by a diagram for a ribbon fusion category [5]. Let us stress, we wish to replace the S2 ∪ S2 byS1 ∪ S1, introducing gapped boundary structures.

To start with, we consider modular ribbon categories, which have a finite number ofisomorphism classes of simple objects, and an invertible matrix sij defined by Hopf links onthe objects i and j. Kirby framed link diagrams are associated to ribbon vertices, and we seehow simple ribbon pictures may give rise to complicated links.

Figure 1. Octonion units on a cube

3. Knots and gravity3.1. Electric magnetic dualityCategorical axioms lie at the foundation of both condensed matter physics and computerscience. It turns out that topological insulators, for instance, are a good intuition for thedyonic mirror [42] that defines holography in the ribbon particle scheme. Every diagram isinterpreted categorically, so that a cube exists in the same sense as an associahedron, which isan axiom for an n-category. On the one hand we have polytopes, and dually we have generalisedstring diagrams. This emphasis is different from 4-categories for geometric quantisation basedon symmetry [43], but we still have to consider arrows in dimensions 0 and 1, which are trivialfor a braided monoidal category. The idea is to associate geometric duality to cohomologicalduality, where the mirror ribbon category occupies the dual dimensions.

The electric charges of the (massless) Standard Model are listed in table 1, as ribbon diagramson three strands, built with Dirac strings. Both massless neutrino helicities occur, but the othermirror braids are not listed. Observe the analogy between color and neutrino mass, as in thecondensate CSFT scheme [22][23]. In the mirror set of diagrams we have the extra magneticdegrees of freedom, so that particles at the mirror carry dyonic charge.

The Chern-Simons action is applied to gravity in 2 + 1 dimensions [44] using the MonsterCFT of central charge 24. In motivic gravity, we build the mass gap for neutrinos, and there isa second CSFT for QCD and the strong CP problem [22][23]. The partition function of [44] isthe modular j-invariant. Recall that this invariant is defined in terms of the Eisenstein formsE4 and E6, where E4 counts roots on the e8 lattice. The j-invariant itself counts irreps for theMonster moonshine module, and its special real values include the golden ratio φ = (1 +

√5)/2

[45].Compare the factor of 32 in (9) with the common normalisation of 1728 = 64 × 27. The

2048 = 64 × 32 equals j(±φ), and the set {±φ,±φ−1} is included in the critical set of realvalues of j. The conjugates of φ solve the quadratic in x which results from insisting on (i) thegeometric sequence Fn+1 = xFn and (ii) the Fibonacci recursion Fn+2 = Fn+1 + Fn. Thus φ isa placeholder for all rationals Fn+1/Fn.

To understand duality [7] we look at combinatorial degrees of freedom. Braids will be relatedto the associahedra. Two copies of e8 have 480 roots in

496 = 2(28 + 28) + 2(8× 8 + 8× 8 + 8× 8) (10)

dimensions, giving 8 copies of 14 in the adjoint part. These are our basis associahedra. Theoctonion factors of 8 will be associated to 3-cubes, so that everything is encoded in dimension 4,since the tensor product corresponds to a sum of basis cube dimensions. More conventionally,the 14 trees label a 14 dimensional theory associated to the 3-time grading [46]

e8(−24) = 14 + 64 + (SO(11, 3) + 1) + 64 + 14. (11)

Here the 64 = 35 + 21 + 7 + 1 counts the ordered subsets of up to three distinct units ine1, · · · , e7, and another 64 counts the subsets of size ≥ 4. That is, we have a 3-brane andmagnetic 7-brane and the usual SO(9, 1) ' SL2(O) embeds in SO(11, 3). As noted above, 7dimensions means 7 ribbon strands, for a total of 14 braid strings. One may extend [46] thisfurther to a (19, 3) theory in the tower of exceptional periodicity. Eventually we find SO(28, 4)breaking to SO(3, 3) × SO(25, 1), where 32 dimensions is high enough to obtain four copies ofthe integers Z8/2, used to define entries in SL2(C).

All higher dimensional associahedra are products of the polytopes in dimension 2 and 3 [47].The three dimensional associahedron of figure 12 is a model for the sheaf cohomology of RP2.As is well known, the associahedra also describe the compactification of the genus zero modulispacesM0,n for Riemann spheres. At very high genus, which is relevant for H2 homology in thesmooth category, the moduli M∞,1 has a completion M+

∞,1 [48] such that

π3(M+∞,1) = Z24 +G, (12)

for some G. Recall the stable group πn+3(Sn) = Z24 for n ≥ 5. Baez [48] describes Z as the

decategorification of a category of tangles, where the objects are strings of n ± signs. Recallthat 24 signs is the setting of the Golay code, underlying the Leech lattice [49][50]. Signs fortangles exist whenever duals are present, which is the case for all our categories.

Below we put braid group generators directly on the vertices of an associahedron. But thereis another deep connection between links and the associahedra, as follows. Given any pair ofrooted, binary trees t1 and t2 on d leaves, there is a pairing h(t1, t2) which defines an elementof Thompson’s F group [51][52]. A traced pairing diagram built from t1 and t2 is a trivalentplanar graph, whose edges may be colored with 3 colors such that each vertex carries one edgeof each color. A ± sign is then attached to each vertex, as in figure 2, depending on whetherthe permutation of (123) is odd or even. The sign determines a link crossing when a trivalentvertex is extended to the crossing piece [52]. It turns out that all links may be obtained thisway.

The four color theorem for planar maps is closely related to these questions, and theconnection between this theorem and pentagons has a very long history. We show a 3-coloringof an associahedron in figure 11.

3.2. Qutrit rest mass eigenvaluesA loop is a quasigroup with an identity, in analogy to a category with a nonassociative productand noncommutative braiding. For example, the integral octonions form a finite nonassociativeloop. Product tables for finite loops are Latin squares. Two simple examples of order 3 are theleft unit loop and the idempotent loop,

Llu =

1 a bb 1 aa b 1

, LI =

1 b ab a 1a 1 b

, (13)

Figure 2. Trivalent vertex maps to link crossing

which are 1-circulant and 2-circulant symmetric, respectively. In general, an order 3 table isselected [53] from 9 points in the Hamming graph H(3, 3), which is the 27 points on the qutrit3-cube (see below) of length 3 words on three letters {1, 2, 3}. For example, the pure state 221puts a 1 in both the second row and second column of Llu.

Hermitian 3× 3 matrices in a generalised Jordan algebra are the natural place for rest masstriplets in the low energy regime. They are necessarily 1-circulants, belonging to a group algebraFS3 on the three object permutations, and diagonalised over C by the quantum Fourier transform(15). Similarly, a real 2 × 2 circulant has basis {1, σX}, where σX is the flip Pauli matrix ofsignature (1,−1). We view either spacetime or momentum space [2] as a six dimensional entity,based on three 2× 2 circulants, because under the action of the Lorentz group cover SL2(C)

Q =

(n mp q

)(a bb a

)(q −m−p q

)=

(t+ z x+ iyx− iy t− z

)(14)

is a vector (t, x, y, z) in Minkowski spacetime, with determinant t2 − x2 − y2 − z2. Clearly itmust be tripled to obtain the full (x, y, z) degrees of freedom.

When our ring R is eight dimensional over Z, as above, three dimensional spaces are secretly24 dimensional. We therefore expect fundamental phases like π/4 and π/6 [2]. These phasesappear automatically in mutually unbiased bases [54][55][56] for qubits and qutrits. In a primepower dimension d = pr there are d + 1 MUBs and d − 1 mutually orthogonal Latin squares,like the pair above for d = 3. More general Gauss sums for modular categories appear in [57],and the connection [58] to Frobenius algebras is discussed below.

Let ω = (−1 +√−3)/2 be the cubed root of unity. The qutrit Fourier transform is given, up

to permutations, by

F3 =1√3

1 1 11 ω ω1 ω ω

. (15)

Its columns form one basis in a set of four MUBs for qutrits. The other 3 bases form a cyclicgroup C3 ⊂ S3, and a cyclic group Cd appears in any prime power dimension d = pr [56]. Thedensity matrices of these columns are the idempotents

B =1

3

1 ω ωω 1 ωω ω 1

, C =1

3

1 ω ωω 1 ωω ω 1

, A =1

3

1 1 11 1 11 1 1

, (16)

and a Hermitian mass operator is a combination of these idempotents. Let

√M = aA+ bB + cC (17)

for a, b, c real. Our masses are the squares of the three eigenvalues of√M , accounting for the

chiral components of our mass states. Without loss of generality, fix a mass scale by the rule(a + b + c)2 = 1. The Koide rule [59][60] follows from the eigenvalues of the charged leptonmatrix

√M =

√µ√

2

√2 θ θ

θ√

2 θ

θ θ√

2

, (18)

where the scale µ = 4/3 follows from (a + b + c) = 1. For charged leptons, the 4 in µ rescalesto the mass of the proton, and the observed value of θ is close to 2/9. The quarks have massmatrices whose phases are 1/3 and 2/3 of this value. The observed neutrino scale is around 0.01eV, and its phases are 2/9± π/12 [2][61].

We embed our Hermitian elements from J3(C) in a higher dimensional exceptional Jordanalgebra [2]. Recall that triality acts on the three off-diagonal copies of O in a 3 × 3 element ofJ3(O), and all circulants belong to a group algebra for the permutations S3, which is our basicHopf algebra [62]. Mass matrices use the cyclic group C3 ⊂ S3, and diagonal mass triplets arefunctions on C3, so that under the Fourier transform the quantum double D(S3) of S3 looks likean algebra FC3 ⊗ FC3, in which electric magnetic duality will become completely transparent.

The Fibonacci B3 representation is 2 × 2, fitting in three ways into a generic 3 × 3 unitarymatrix. A circulant mixing factor is automatically in SU(2)×U(1), and the product of F3 andthe real form of the tribimaximal matrix gives a 3×3 representation [61] of the arithmetic phaseπ/12. Under Fourier supersymmetry [1], neutrinos are mapped to the photon identity braid,while e± 7→ W±. The Z boson is associated to the choice of twist for the Fourier transform.There is no need for any particles beyond the SM, and the ±π/12 phase is analogous to thephase for the weight 1/2 Dedekind η function. Fibonacci categories have doubles, which are anatural setting for our mirror pairs.

Table 1. Standard Model electric braid states

L R (1)L/R (2)L/R (3)L/R

ν σ1σ−12 σ−12 σ1 m1/N1 m2/N2 m3/N3

ν σ−11 σ2 σ2σ−11 N1/m1 N2/m2 N3/m3

e− (−−−) (−−−)e+ (+ + +) (+ + +)u (−− 0) (0−−) (−0−)d (−00) (0− 0) (0−−)u (+ + 0) (0 + +) (+0+)

d (+00) (0 + 0) (00+)

To give the local charged leptons mass through emergent Yukawa couplings, we need a(GUT) quark neutrino complementarity, swapping color and chirality, explaining why chirality isrepresented in B3. The Fourier transform is associated to geometric duality in our 4-categories,where a braided monoidal representation category has 2-arrows, 3-arrows and four dimensionalaxioms.

Under chiral symmetry breaking, fermion pairs create the Bose-Einstein condensates of thequantum vacuum. Our central right handed neutrino mass corresponds to the present dayCMB temperature at 0.00117 eV, indicating that the CMB is direct evidence for a condensatecosmology.

4. The categorical perspective4.1. Beyond set theoryAxiomatically, quantum gravity is about categorical logic for propositions about the quantumvacuum, whose structure begins with the cosmological neutrino ansatz [1][2]. Recall that classicallogic employs sets and distributive lattices, where Stone’s theorem [63] states that the spaceassociated to a lattice is Hausdorff if and only if the lattice is Boolean, defining a category ofStone spaces, which is a special limit of the category of finite sets. Ordered Stone spaces areessentially coherent spaces, and coherent locales are essentially locales of ideals in a distributivelattice. Distributive lattices are Boolean only if all prime ideals are maximal. In short, classicalspaces are derived from lattice algebras with a number theoretic flavour.

Quantum mechanics immediately requires nondistributive lattices, and axioms for higherdimensional categories [64][65]. The polytopes of scattering theory [65] associate particle numberwith dimension, naturally introducing infinite dimensional categories, starting with the 1-operadof the associahedra. The category of Hilbert spaces for quantum mechanics is a symmetricmonoidal category, but for gravity we permit a non trivial braiding.

Replacing the Boolean truth values {0, 1} with R takes us from Stone duality to either Gelfandduality (for commutative rings) or R/Z = S1 in Pontryagin duality. But we need not give S1

a real structure immediately when algebraic number fields are in play, so long as we note thatS1 should contain a copy of every cyclic group. Quantum mechanical propositions localise todefinite rational prime powers, like p = 22 for two qubits. Only a maximal category of allpossible state spaces would require a notion of real number. Thus our braid loops are not at allS1 spaces in the usual sense.

For nonperturbative structures, we need a monadic connection between algebra and geometry,defining endofunctors on true categories of motives. If a ring R is commutative, its set ofidempotents forms a Boolean algebra, and any commutative R is a ring of global sections fora sheaf on a Stone space [63]. The canonical such sheaf is the Pierce sheaf, based on theStone space spec I(R), where I(R) are the idempotents of R. Pierce decompositions extend tononcommutative and nonassociative algebras based on H and O. In particular, the integral partof the exceptional Jordan algebra J3(O) plays a key role in motivic gravity [2][46][66].

Nondistributive lattices in ordinary quantum mechanics are usually commutative, because forvector spaces the unions V ∧W are commutative. Tensor products are also weakly commutativein the symmetric monoidal structure, just as Cartesian products are for sets. Our braidingsbreak this symmetry, and this characterises the particle spectrum of the Standard Model.Distributivity is discussed further in the last section.

4.2. Frobenius and Hopf algebrasMutually unbiased bases [54][55][56] and related observables in ordinary quantum mechanicsare associated to algebra objects in a symmetric monoidal category. A pair of complementaryobservables [58] determines a pair of interlaced Frobenius algebras, which form Hopf algebras asfollows.

A dagger symmetric monoidal category has an involutive functor Co → C which equals theidentity for objects, and an arrow f is self-adjoint if f † = f . In a strict monoidal category C,we consider the subcategory P generated under ⊗ by a single object. For example, take allqubit spaces in the category of finite dimensional Hilbert spaces. Since the objects in such asubcategory are labelled by N, it is called a PRO. A strict monoidal functor P → C will be calleda P-algebra (the usual term is T -algebra, but this has multiple meanings for us). If there is asymmetric group action, a PRO is known as a PROP.

Now consider a PROP for commutative monoids, generated by an object P . There aremultiplications µ : P ⊗ P → P and unit maps η : 1 → P , where as usual µ is depicted as atrivalent vertex and associativity holds. Anything can be weakened in higher dimensions, but

this is a reasonable setting for quantum mechanics. The cocommutative comonoids have upsidedown diagrams, with a coproduct δ : P → P ⊗ P and counit ε : P → 1.

We care about the PROP [58] for commutative Frobenius algebras. These are bialgebras(µ, η, δ, ε) such that string duality holds on the 4-valent diagrams δµ : P ⊗ P → P ⊗ P . If it isalso the case that µδ = 1, the algebra is special. For the special commutative Fronenius algebrasin a dagger category, the spider theorem [67] says that all tree components (both upward anddownward pointing arrows Pn → Pm) collapse to a single vertex, so that the fusion trees areunimportant. This PROP is equivalent to the category of cospans on finite sets.

It turns out that we have a dagger special commutative Frobenius algebra exactly when anorthonormal basis {pi} for P satisfies

δ1 : pi 7→ pi ⊗ pi, ε1 : pi → 1 (19)

for all i. That is, a basis vector is grouplike for the coproduct, meaning that it is copied like aclassical operation. Our quantum monad for gravity is motivated by this concrete descriptionof measurement, which singles out set like objects in a basis. For a Hilbert space of dimensionn, let i, j ∈ 0, 1, · · · , n− 1. Then there is an algebra with µ2 : pi⊗ pj 7→ pi + pj and η2 : p0 → 1.It is only weakly special, as µδ = n · 1, bringing in the normalisation factors for MUBs.

Another important coproduct, written here for a finite abelian group, is δ2 : g 7→∑

a+b=g a⊗b.The pair (µ2, δ2) form a Frobenius algebra. For quantum complementarity, the trick is to mixup the algebras and coalgebras on a Frobenius diagram, because with a Fourier transform, weare interested in interacting observables. So we color the µ vertex differently from the δ vertex,and the two mixed algebras (µ1, δ2) and (µ2, δ1) are Hopf algebras. For example, for a finiteabelian group, (µ2, δ1) is the group algebra and (µ1, δ2) applies to group characters. Coproductsof the form δ2 underlie the construction of Jordan algebra pairs. In these commutative Hopfalgebras, the antipode always satisfies S2 = 1.

For condensation in gapped boundary systems, where fusion becomes important, we permit aweakening of such structures, but the algebra object remains commutative, as we would expectfor an observable that manifests itself in classical reality.

4.3. The pentagon of treesLet us return to the basics of polytopes and braids. A finite dimensional module over a ringR typically has a basis set. For example, figure 1 is the lattice of subsets for a three elementset {I, J,K}, which form a basis for space. Here we have reduced the 8 dimensions of O to a 3dimensional object, whose seven non trivial units give the Fano plane. Given a three elementset {I, J,K}, its subsets are generated by the polynomial

(x+ I)(x+ J)(x+K), (20)

and similarly for any n point set. Setting I = J = K = 1 recovers the binomial coefficents, whichgeneralise to the Gaussian polynomials when I = 1, J = t2, K = t−2. For four variables, theGaussian polynomials come from {t−3, t−1, t, t3}, and so on. Thus polynomials in more variablesmay be obtained when I, J and K are not fixed in the usual fashion.

For quantum logic, elements are initially lines, rather than points. In figure 3, we replacewords by line configurations. Each letter represents an intersection point, so if we look at theintersection points on the lines, the double letters (IJ etc.) disappear from the cube, leaving a5-cell of five points. Such 5-cells often appear in higher dimensional lattices.

A planar projection of a 5-cell is a pentagon. The pentagon of figure 4 carries a variety oflabellings. As the first polytope in the sequence of associahedra, it’s vertices are the binary rootedtrees with five leaves, including the root. The noncommutative forests are easily derived from

Figure 3. 5-cell from three lines

Figure 4. The pentagon vertices as (i) binary rooted trees (purple) (ii) noncommutative forests(brown) (iii) non-crossing partitions (orange)

the trees by looking at the areas between the tree edges. These labels exist for the associahedronin any dimension.

Another natural labelling of the pentagon uses elements of the braid group B3, which hasgenerators σI and σJ satisfying the group law

σIσJσI = σJσIσJ . (21)

Observe how the blue words on the pentagon match the non-crossing partitions of the dottriangle, when the vertices on the triangle are labelled I, J,K. Now we use the letters I, Jand K to represent elements of B3 and include other elements of B3 to cover all vertices of thepentagon. A non-crossing partition indexes a braid [68][69] when the partition is assigned apermutation in S3, such that the identity 1 is the source of the pentagon, as shown. Given 3points in a disc, the permutation looks at the triangle defined by the 3 points and says wherethe braid will send each point around the triangle.

In this way, the braid group Bn in any dimension is mapped to the vertices of theassociahedron in dimension n − 1, and the generators of Bn are mapped to initial directionson the polytope. Our ribbon charges correspond to vertices on the cubes, and we combine allrelevant polytopes in a higher dimensional operad for ribbon diagrams.

4.4. Fibonacci braids and condensationAn example of a cyclic B3 representation in H is [70]

σ12 =1√2

(1 + i), σ23 =1√2

(1 + j), σ13 =1√2

(1 + k). (22)

The pentagon also includes the identity 1 and the product σ12σ23σ13. Under the Pauli matrixrepresentation for H we have

σ12σ23σ13 =i√2

(1 11 −1

), (23)

which is the Fourier transform. A rotation of this representation in SU(2) takes us to theFibonacci anyon representation, which is 2 × 2 for B3. Consider now the closely related 3 × 3cyclotomic representation of the four strand braid group B4 in [71], namely

σ1 =

e3πi/5 + φe−3π/5 0 0

0 e3πi/5 0

0 0 e3πi/5

, (24)

σ2 =

e3πi/5 + φ−1e−3πi/5 0 φ−1/2e−3πi/5

0 e3πi/5 0

φ−1/2e−3πi/5 0 e3πi/5 + e−3πi/5

,

σ3 =

e3πi/5 0 0

0 e3πi/5 + φ−1e−3πi/5 φ−1/2e−3πi/5

0 φ−1/2e−3πi/5 e3πi/5 + e−3πi/5

.

A similar representation exists for all Bn with n ≥ 3 in a dimension given by the correspondingFibonacci number, and is universal for quantum computation. The qutrit components for theFibonacci anyon are labeled by the words IX, XI and II, where I and X are the two objectsand X ⊗X ' I +X is the non trivial fusion rule.

Thinking of dissipation for thermal gravity, note that Fibonacci fusion is an example ofnear-group fusion [72] on a not necessarily invertible object X, namely

X ⊗X ' G+ kX, (25)

for a group G and ordinal k. For |G| = k+1, the category exists only when G is the multiplicativepart of a finite field, the cyclic group Cpr−1. Examples of interest include (G, k) = (C2, 1), whichhas three ⊗ structures [72], and (Cp−1, p − 1) for a prime p, which defines a sequence Fibp ofFibonacci categories, starting at p = 2. At p = 3 we obtain the rule

X ⊗X ' I1 + I2 + 2X, (26)

where we write I1 and I2 for the objects in G. This is known as the e6/2 rule. The prime pcorresponds to the qudit points on a discrete cube, where the usual parity cubes give qubit stateson 0 and 1. Targets of parity cubes also represent square free ordinals. The trit at p = 3 gives adivided cube whose dimension is fixed by the number of X letters in a word. For example, the9 point square holds all words with only one X, such as I1XI1 at 11 or I2X at 20.

Hopf algebras graded by G generalise supersymmetry [73], which we see here at p = 3. Herethe object I1 + (I1 + I2) in the category of vector spaces has a unique C2-graded Hopf structurewhich is a quotient of C[x, y] with an antipode S(x) = x and S(y) = −y. The category of C2

graded vector spaces is thought of as a condensation of the modules for the Hopf structure,which happen to give the category of representations of the permutation group S3. Recall thatthe double D(S3) governs electric magnetic duality, as discussed below.

The ordinary Fibonacci category Fib2 is a condensation of its double category, with specialobject 2I +X [73]. In this case we have an unconventional antipode satisfying S10 = 1, where

S = 1I1 + 1I2 −φ−1 + ρi

21X , (27)

with ρ =√φ+ 2.

By definition [73], an algebra object A in a braided fusion category is condensable if (i)mσUV = m is commutative (ii) Hom(1, A) is equivalent to the underlying field, and (iii) themultiplication m comes with a splitting map A → A ⊗ A. Compare this to the commutativeFrobenius algebras. The only new thing is connectedness, condition (ii), which is true for Hilbertspaces by linearity. A condensation functor C → CA has a right adjoint, and the compositionof adjoints is a Hopf comonad. Thus condensation is the correct setting for the quantisation ofclassical monads, such as the power set monad for classical logic.

A monad T with its structure map T 2 → T is the ultimate generalisation of an idempotentfor measurement. Abstract condensation is encoding the collapse of the wave function. Observehow the Fibonacci fusion rule X ⊗X ' X ⊕ I may be interpreted: as a projective idempotentof the form X2 = IX.

Now consider ribbon representations, starting with representations for the quantum doubleof a finite group. These categories arise with the condensation of anyons to a surface boundary.A hexagonal lattice is considered in [7], but the arguments apply to any lattice. The Dijkgraaf-Witten model in [9] uses an inner and outer rectangle of plaquettes in a square lattice for adiscrete G = S3 gauge theory, with an initial Kitaev Hamiltonian. Here a ribbon is a chain ofsimplices in the lattice, and each simplex carries a qudit. The ribbon operators on the latticeform a Hopf algebra dual to D(G).

Gapped boundary types correspond to subgroups of G, such as our C3 ⊂ S3. Recall thatD(C3) uses the Fourier transform F3 to make electric magnetic duality manifest [62]. Asexplained in [7], the trivial subgroup gives electric charge condensation, while the fullgroupG gives flux condensation. Elementary excitations in general, here for a finite group G [9], aredyonic pairs (m, e), where the magnetic charge m is a conjugacy class for G and the electriccharge e is an irrep for its centraliser. In other words, a dyon is an irrep for the quantum double.For example, m = {(231), (312)} and e = C3. This explains the choice of particle braids in table1.

Anyon fusion occurs when two excitations are brought to the same simplex on the lattice.Compare this to figure 7, where two types of overlap triangle are possible. In defects on theboundary, these two options define two distinct tensor products. A gapped boundary in [9] is acondensable algebra object A, in a unitary modular tensor category, which is also Lagrangian,meaning that the quantum dimension of A is the square root of the full category dimension.

A collection of n gapped boundaries (internal rectangles on the lattice) models n markedpoints on a Riemann sphere, and hence n anyons for the fusion trees on the associahedron withn leaves. A sequence of splittings from the vacuum object, followed by condensation of n objectsto the vacuum, is exactly a choice of two trees on the associahedron, which defines an elementof Thompson’s F group, as described in section 3.1. These are ground state degeneracies. Thustwo boundaries A1 and A2, along with the vacuum, define a pair of pants diagram, and in thiscase a bulk anyon particle/antiparticle pair (which condenses) may be represented by a (Wilson)line connecting the two holes on a surface.

Condensation introduces 3j symbols into the structure [7] of the category, generalising trialityon V1 ⊗ V2 ⊗ V3. Its equation compares two diagrams: one that first fuses two bulk anyons andthen condenses them onto a boundary, and one that condenses two anyons separately. Thesymbols have six indices: three for the 2 + 1 bulk anyons and three for the condensation verticeson both diagrams, where the latter are basis indices for the Vi. The required axiom is a pentagonwith four 3j arrows and one fusion arrow [9], and the commutativity of condensation relates the3j to the braiding operator. Finally, a 6j symbol is defined across the boundary by a mirrorpair of 3j equations, summing over both the input bulk fusion and the mirror fusion index. In[7], a trivalent vertex is built using a 3j symbol, but our boundaries [9] make 4-valent graphs,which turn into associator trees on the selection of a root leaf. Thus the pentagon is essentially

Mac Lane.Our 3 × 3 algebras for B4 are augmented by MUBs for Dirac operators, including a 4 × 4

Fourier MUB for the γ5 matrix. This now appears in the Lie algebraic triality automorphism τfor D4 in terms of the modular S and T matrices for the toric code fusion category, for whichG = S2 in the above. That is,

−2τ = ((41)(32)) ◦ S ◦ T =

0 0 0 10 0 1 00 1 0 01 0 0 0

1 1 1 11 1 −1 −11 −1 1 −11 −1 −1 1

1 0 0 00 1 0 00 0 1 00 0 0 −1

. (28)

The 4-basis is {1, e,m, em} [9], and the Lagrangian algebras are A1 = 1 + e and A2 = 1 + m.Flux condensation sends 1 and e to 1, and m and em to m, and the braiding moves one holearound another.

The usual Fibonacci category is a condensation of its double. The representation of Bn in(24) comes with a ribbon twist map θX = −e±πi/51X . We assume that the

√φ appears as

a Lagrangian dimension for some higher dimensional structure, and the antipode of (27) alsorequires rings with the number ρ, as expected. Recall that the scale factors (1, φ) are associatedto one copy of e8, introducing a (4, 4) metric under the negative norms involving φ.

Consider twisted doubles. For the cyclic group C2 of the toric code, it is shown in [74] thatone 3-cocycle C2 × C2 × C2 → U(1) takes the value −1 on the target (111) of the parity cube,and the value 1 elsewhere. The ground state degeneracy is 4, and these signs determine the2 × 2 Fourier transform. This defines a twisted double lattice model, whose string dual is theLevin-Wen model for the quantum group Uq(sl2(C)) when q = ω. Recall that q = ω has twosimple reps, while the category at the fifth root of unity has four. For C3, there is only onetwisted double with a Levin-Wen dual, whose cocycle defines the nine entries of the Fouriermatrix F3. But there are two other cocycles [74] for C3 with a ground state degeneracy of 9,including one based on the topological phase θ = e2πi/9.

Recall the planar pants diagram, with a bulk anyon line connecting the two interior circles.A pair of S1 circles is a toric analog of the pair of S2 which form a 1-handle in a Kirby diagram.Thus a single bulk anyon ribbon is a natural surface analog of the cancelling 2-handle, andfor both T 2 and S2 × S2 the 2 × 2 intersection form is given by σX in (6). The torus is thecompactification of the (1, 1) Minkowski plane, while S3×S3 ⊂ R8 compactifies our (3, 3) space.

Thus we interpret the three copies of σX in (7) as three gapped boundaries for massgeneration. Then for D(C3) we have two condensates: 1 + e+ e and 1 +m+m. This category

is the same as SU(3)1 × SU(3)1, which we can represent with two surface layers, so that thebulk line between two holes is categorified to a cylinder handle connecting the holes on differentsheets. Sliding tubes past each other in this 3-space is like wormhole braiding, and the dimensionof the theory has been raised. For one cylinder, say connecting an e+e− pair, we consider aribbon strip in a CFT as a boundary of the bulk cylinder.

Given a two dimensional cylinder of radius r, which is CP1/{0, 1}, we define a CFT energymomentum tensor for a periodic strip of width 2πr in the plane [75]. The anomalous

〈T 〉 =c

24r2(29)

is due to the finite size, as in a Casimir effect for the two edges, thought of as horizons in oppositedirections. When the two horizons are free, this is reduced by a factor of 4. In dimension four,quantum inertia [76][77] is a Casimir effect for two celestial horizons [2]: the local Rindler horizonand the cosmological horizon of the EW vacuum. For very low accelerations the rest mass isreduced, and it becomes zero at an Unruh wavelength of 4R for R = πr the Hubble radius.

The factors of π that often appear (in comparing circle clocks to radial clocks) arehypothesised to be responsible for an exact Koide 2/9 parameter. For instance, in 2/9 =(2π/27)(3/π) the second factor is an exact effective dark mass-energy fraction, derived fromthe Friedmann equation using a semiclassical pair production argument [78]. Since we areworking beneath classical equations, the 2/9 might arise fundamentally as (2π/9)(1/π), withthe 2π/9 appearing above as the cocycle for the C3 twisted double [74]. This deserves furtherinvestigation, especially since the critical strip of the Riemann zeta function ζ(s) has an inertialline at s = 1/2.

4.5. Templates and ribbonsCategorification is inevitable in quantum computation, where lines are thickened to ribbon strips,used to build surfaces with boundaries. A template is a branched surface which includes ribbonvertices, as shown in figures 5 and 6. In 1995, Ghrist [79] showed that there exists a templatewith four holes containing all knots and links, as one would expect for a DNA code.

Figure 5. A two-holed Lorenz attractor with blue path

Everything happens in either three or six dimensions, because the higher dimensions ofextended M theory just correspond to extra strands in our braid diagrams. This has beendiscussed elsewhere. Figure 6 illustrates the standard product and coproduct diagrams for abialgebra, reading the processes down the page. Templates [80] also include up and down ribboncaps for duality, as in the Lorenz template of figure 5.

Figure 6. Template vertices

A template diagram has a framed link equivalent, as shown in figure 7. Kirby moves act onthe framed links.

For the B3 diagrams, there exists a universal representation of SU(2) using the Fibonaccianyons [71][81]. Our Standard Model particle braids [82][65][38][39] assign SU(3) color to achoice of one in three twisted ribbon strands, and the twist is a U(1) charge, as in table 1.

Figure 7. Framed links for template vertices

4.6. Simplices and polytopesA discrete cube is a finite piece of the integral lattice with d points along each edge segment.Its vertex coordinates are noncommutative words in the integer letters. For commutativecoordinates, we take diagonal slices. For example, the two words (10) and (01) sit at eitherend of a diagonal line across the square in the plane. We replace letters with the variables Xand Y , so that the integers count the number of appearances of X and Y in a word. Then Xand Y are directions in space. For a three letter qutrit alphabet, we get 2-simplices, as in theexamples of figure 8.

Figure 8. Discrete simplices on three letters

A pentagon has natural integer coordinates [83], as in figure 9, so that three pentagons sitinside the tetractys simplex on the right. Observe the correspondence between these coordinatesand words on the pentagon of figure 4. We can do this for the associahedron in every dimension,using discrete simplices with d+ 1 points on each edge.

Figure 9. Coordinates for a pentagon

The associahedron of figure 11 has 14 vertices, and 24 triangular faces when each of its sixpentagons is divided into three triangles. When all faces are triangulated, the associahedron isdual to the 24 vertex permutohedron for S4. Coordinates for the associahedron are extended[1] to the 120 vertex polytope of figure 10, which is a pentagon blow up of the permutohedron.Two copies of this polytope catalog the 240 roots of e8 in dimension 4, where the scaling factors(1, φ) come from the icosian integers. The e8 roots in the magic plane attach Jordan algebraelements to the six points of the star [66], lying inside the six points of its a2 plane. These arethe 12 points of g2 coming from the vertices of the cuboctahedron on the three qutrit cube.

Again, the 24 vertices of the permutohedron are each blown up in the 120 vertexpermutoassociahedron of figure 10. Let us divide the 24 pentagons into two sets, green and red.The green lines connecting the 12 green pentagons form the icosahedron, with 20 triangularfaces.

Figure 10. Icosahedron inside the permutoassociahedron

The 12 vertices of the icosahedron have traditional coordinates of the form (0,±1,±φ), withcyclic permutations. The 6 lines through a centred pentagon on the icosahedron come from a6 dimensional lattice. In figure 10, the 8 green triangles inside a 12-gon form a square on 8out of 20 vertices of the dodecahedron. These are the vertices (±1,±1,±1), where the other 12coordinates are cycles of (0,±φ,±(φ− 1)). The 12 vertices of the icosahedron are similar to the12 vertices of the cuboctahedron, inscribed on the 12 edges of a cube.

The golden number ρ appears in the right angled triangle with an angle of 36◦ and sidelengths (

√5, 2ρ, φρ). A pentagram component is the right angled triangle (1,

√φ, φ). To include

1/√φ, as in (24), we need integers of degree 8, or 16 with the complexification of (27).

A discrete direction in our computational space is labelled by the toric paths 1, X, XX,and so on. Given the qudit interpretation, we want an auxilliary space whose directions aregiven by prime powers, so that all qubit cubes are given by a discrete edge, as in the sides oftriangles in figure 8. The figure 8 simplices then belong to higher dimensional qutrit cubes,where the dimension is determined by the number of letters in the word labelling a point. Thusas usual the commutative tetractys diagram gives the 27 points on the 3-cube. The central wordXY Z holds 6 permutations for the six paths on the little cube with target point XY Z. But indimension 4, the 81 path 2-simplex now labels points on a 4-cube. So we can either increase thenumber of points along an edge in dimension p, and take the diagonal simplex, or we can fix ppoints on an edge and increase the dimension. In the former case, the dimension is constrainedby the qudits, and four dimensions carries 4-simplices for 5-dits, while two dimensions carriesthe qutrits.

Introducing the prime 7 in dimension six, we get the cubicuboctahedron and the Mathieu

Figure 11. 3-coloring of the associahedron

group M24 [84], starting with a permutoassociahedron model for the genus 3 surface. The sevenprimes (including 1) p that divide the order of M24 are precisely those such that p+1 is a divisorof 24. M24 has 26 irreps, and their dimensions satisfy nice properties. Only the largest irrep, atdimension 10395, has a new prime factor, namely the 24th prime 83.

5. Motivic pairingsPROP categories rely on a higher dimensional notion of distributivity [58][85]. For quantumlogic, where set cardinality is replaced by dimension, the reals are automatically infinitedimensional, and we would like to think of distributivity in ∞-categories. Here however, itdepends on a braiding between ⊗ and ⊕ structures.

Observe that while all n ∈ N have a unique prime factorisation, all n ∈ N also have a uniquesum decomposition

∑i Fi into non consecutive Fibonacci numbers. So we think of the ∞-

category distributivity in terms of maps between the product and sum representations. Morally,this canonical category of motives underlies L-functions, like the Riemann zeta function. TheBi matrix representations under ⊕ have increasing dimensions Fi inside Bn, while there is astate space of dimension n = pr11 · · · p

rkk under ⊗.

Quantum distributivity fills in higher dimensional cells. The union of lines I and J is theplane IJ , suggesting a square face on a cube, or pseudonatural transformation. In our double4-category, 2-arrows are objects in two different ways.

All computations are motivic. An integral is generically a pairing between universal homologyand cohomology. The isomorphism between these spaces is natural when objects have both ageometric and algebraic interpretation, wherein our topological field theories become monadicendofunctors. Such a pairing is generically a functor F : Co × C → R, so that a map betweentwo such functors F and G is a dinatural transformation α [86], which satisfies the hexagonalrule

F (D,C)→ F (C,C)→αC G(C,C)→ G(C,D) = (30)

F (D,C)→ F (D,D)→αD G(D,D)→ G(C,D)

for every f : C → D in C. So it’s basically a natural transformation for the spans and cospanson an original category of sets. Given such a functor F , a coend of F is a pair (C,α), with Can object and α a dinatural transformation F → C that maps F to a constant.

Let C be a coend in a ribbon category C. A Kirby element in a ribbon category [87] isa morphism f : 1 → C such that any framed link L on n strands defines a good invariantT (L, f) = aL ◦ f⊗n, where aL : C⊗n → 1 is the unique arrow that attaches the link to an object

X1⊗ · · · ⊗Xn. In a ribbon fusion category, with a finite set s of simple objects, a coend has theform C = ⊕i∈sX∗i ⊗Xi. In the double Fibonacci category Fib2, this gives us the special object2I +X.

Under the Thompson group construction of section 3.1, a fusion vertex can become a braidcrossing on a link. But we can never obtain a 3-coloring at a vertex in the Fibonacci category.In a category where we can have three colors at a vertex, like an annihilation (e, e, 1) interactionvertex, the bulk fusions in a ground state 1→ 1 diagram are removed, and there are no verticesin the resulting link diagram. If the initial crossings lie inside a set of holes on the link diagram,the hole boundaries are then connected by non crossing lines, as for planar algebra diagrams.We see then that these holes add structure to the local cutouts used in skein relations.

Finally, figure 12 is a heuristic view of a proper helicity neutrino at the Thompson scatteringmirror, with a collapsed RH braid representing the sterile state of the CMB.

The moral of the story is that motivic geometry cares about numbers. We do not start withmessy real or complex analysis, or classical gauge groups, since these methods rightly exist onlyas a limit of local (meaning at a prime) computational diagrams.

Figure 12. Neutrino braid at mirror with RH line

AcknowledgmentsThe author thanks Vaughan Jones for speaking about the Thompson group trees in Auckland,some time in 2016.

Appendix A. Number TheoryA good introduction to Number Theory is [88]. We introduce a selection of interesting fieldsand rings with a mind to applications in quantum computation and quantum gravity.

By definition, a number field extends the rationals by one special real number α, containingall numbers of the form a+ bα for a and b in Q. Multiplication and addition in this field Q(α)work in the obvious way. Given α, there is a ring of integers in Q(α), but this is not always theinteger multiples of the form a+ bα. For example, when α =

√5, the ring of integers consists of

numbers a+ bφ, where φ = (1 +√

5)/2 is the golden ratio.An algebraic number is a root of a finite polynomial with rational coefficients, such that the

leading coefficient is 1. The golden ratio is algebraic as a root of X2 − X − 1 = 0. Given analgebraic number, there exists a unique such polynomial (of a given degree) with α as a root. Itis useful to factorise the polynomial. Consider the quadratic X2 − k = (X +

√k)(X −

√k) = 0

of degree 2. The set of roots {+√k,−√k} are called conjugates for the field Q(

√k), and we use

this term for polynomials of any degree d. In terms of the d conjugates, where α = α1, the normof a number α is defined by

N(α) =

d∏i=1

αi. (A.1)

Thus N(φ) = φ · (−1/φ) = −1, while X2 + 3 = 0 gives N(√−3) = 3.

Given a set of conjugates for α, any other element β in Q(α) may be written in the form

β = a0 + a1α+ a2α2 + · · ·+ ad−1α

d−1, (A.2)

where the ai are rational or integer as required. The field conjugates of β are the d− 1 numbersof the form a0αi + · · ·+ ad−1α

d−1i .

Take a basis {β1, β2, · · · , βd} of Q(α). Typically, we will choose the basis {1, α, α2, · · · , αd−1}.Now for any Q(α), the basis defines a d× d matrix Mij with columns indexed by the basis androws by the conjugates of α. Many examples are given below. The discriminant of Q(α) isdefined by the determinant square ∆ = Det(M)2.

Appendix A.1. Quadratic fieldsThe degree d is a quantum dimension, since 2 × 2 matrices ought to be about qubits. Whenα =

√k for an integer k with no square factors, the polynomial X2 − k = 0 defines the field

matrix

M =

(1√k

1 −√k

)(A.3)

with discriminant ∆ = 4k. But for√−3, the ring of integers has a basis {1, (−1 +

√−3)/2},

so that ∆ = −3. Similarly, other negative values of k give negative integral discriminants, incontrast to the positive example of Q(

√5), which has ∆ = 5 coming from(

1 φ1 −1/φ

). (A.4)

There are no additional field conjugates in the quadratic case. The opposite sign in N(φ) = −1,compared to an ordinary complex norm, is responsible for time in a Lorentzian metric, and3+5 = 8 dimensions are associated to the adjoint representation of SU(3) and octonion algebras.

Appendix A.2. Fields on cube rootsNote that the signs in (A.4) give the 2× 2 Hadamard matrix. In dimension 3, we see the qutrit3 × 3 Fourier transform in the field matrix. Let ω = (−1 +

√−3)/2 be the cube root of unity

above. For the polynomial X3 − k = 0 with k cube free, we have

M =

1 k1/3 k2/3

1 ωk1/3 ωk2/3

1 ωk1/3 ωk2/3

. (A.5)

Hopefully it is clear that 27 is an important number! This is the determinant square of theFourier transform, and we have in general ∆ = −27k2.

An elliptic curve C of genus 1 takes the standard form Y 2 = X3 + aX + b for rationalcoefficients, and has a cubic discriminant ∆C = 4a3 + 27b2, generalising the example above. Ifa prime p divides ∆C , then ∆C = 0 in the finite field Fp. The finite set of Fp solutions to Cdefines a Mordell-Weil group for C, whose order Np(C) appears in the zeta function for C.

Appendix A.3. Fields on fourth rootsWhen α is a fourth root, we find that ∆ = −k3 for X4 − k = 0. The matrix is

1 k1/4 k1/2 k3/4

1 ik1/4 −k1/2 −ik3/41 −k1/4 k1/2 −k3/41 −ik1/4 −k1/2 ik3/4

. (A.6)

Let ρ =√φ+ 2 be the diagonal of the golden rectangle. It is algebraic because ρ4−5ρ2+5 = 0.

There are two nice bases for the integers in Q(ρ), namely1 ρ ρ2 ρ3

1 −ρ ρ2 −ρ31√

5 5 5√

5

1 −√

5 5 −5√

5

and

1 ρ φ ρφ1 −ρ φ −ρφ1√

5 3 3√

5

1 −√

5 3 −3√

5

, (A.7)

both with ∆ = 1055.7281. The second basis forms the golden ring of integers of the form

x0 + x1φ+ x2ρ+ x3ρφ (A.8)

for xi ∈ Z. Note that√

5 = 2φ − 1. In the complex field Q(ρ, i), the ring of integers defines adense map of Z8 into C.

Figure A1. Angle 36◦ bisection

Figure A1 indicates one of ten possible blue rectangles covering much of the pentagram. The10 external blue points define a decagon. The chord length on a unit side decagon is ρ.

Appendix A.4. Fields on fifth and higher rootsUsing the golden phase e2πi/5, the discriminant for X5− k = 0 is ∆ = 55k4. For example, whenk = 4 we have ∆ = 800000. We are interested in prime dimensions d for qudit computationspaces. Let θ be the primitive d-th root of unity. For d prime, the phase coefficients θij fori, j ∈ {0, 1, · · · , d− 1} always define the discrete Fourier transform. ∆ for Xp − k = 0 looks likeppkp−1.

Appendix A.5. Primes and latticesFor quadratic fields, there is a quadratic form that characterises the norm. As expected, theform f = aX2 + bXY + cY 2 comes with the discriminant ∆f = b2− 4ac. In particular, for Z[ω]in Q(

√−3) we have ∆f = −3 from X2 + XY + Y 2. For Z[φ] the form is X2 + XY − Y 2 and

∆f = 5. The form X2 +Y 2 matches the Gaussian integers Z[i]. Here Q(√

5) requires φ because5 equals +1 mod 4. For positive primes p ≥ 7 that equal 3 mod 4, such as 7 and 11, the integershave the simple basis {1,√p}, while −7 = 1 mod 4 uses (−1 +

√−7)/2.

The Galois primes {2, 3, 5, 7, 11} give the angles for Lie algebra root systems, which satisfythe lattice condition

4 cos2(2π

p+ 1) ∈ {0, 1, 2, 3}. (A.9)

Beyond Lie algebras there are other important lattices, notably the Leech lattice in dimension24 [50].

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